Cyclicity Preserving Operators on Spaces of Analytic Functions in $${\mathbb {C}}^n$$

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show operators preserve shift-cyclic are necessarily weighted composition operators. Examples for which this result holds true consist the Hardy space $H^p(\mathbb{D}^n) \, (0 < p \infty)$, Drury-Arveson $\mathcal{H}^2_n$, and Dirichlet-type $\mathcal{D}_{\alpha} (\alpha \in \mathbb{R})$. We focus when $1 \leq \infty$, converse is also true. The techniques used to prove main enable us a version Gleason-Kahane-\.Zelazko theorem partially multiplicative linear functionals more than one variable.